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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 14th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItex[[Hopf fibration]]

   Hopf fibration

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 14th 2011
   • (edited Dec 14th 2011)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI added some material to [[Hopf fibration]], some of it in a speculative vein. In this case, I was an Anonymous Coward because I hadn't noticed I was not logged in.

   I added some material to Hopf fibration, some of it in a speculative vein. In this case, I was an Anonymous Coward because I hadn’t noticed I was not logged in.

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeDec 14th 2011
   • (edited Dec 14th 2011)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI just checked that you had not spammed :-) I found some typos as well. (I fixed them.)

   I just checked that you had not spammed :-) I found some typos as well. (I fixed them.)

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 20th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have cross-linked the material at _[[Hopf construction]]_ and _[[Hopf fibration]]_ a little more.

   I have cross-linked the material at Hopf construction and Hopf fibration a little more.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 11th 2019
   • (edited Feb 11th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexbriefly added mentioning ([here](https://ncatlab.org/nlab/show/Hopf+fibration#RealizationViaQuaternions)) of the alternative construction via quaternions that came up in discussion in another thread ([here](https://nforum.ncatlab.org/discussion/9297/equivariant-cohomotopy/?Focus=75960#Comment_75960)) (I am not happy with the structure of this entry, there is much room for making it more comprehensive and systematic, but I'll leave it as is, for the time being) <a href="https://ncatlab.org/nlab/revision/diff/Hopf+fibration/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hopf+fibration/16">v16</a>, <a href="https://ncatlab.org/nlab/show/Hopf+fibration">current</a>

   briefly added mentioning (here) of the alternative construction via quaternions that came up in discussion in another thread (here)

   (I am not happy with the structure of this entry, there is much room for making it more comprehensive and systematic, but I’ll leave it as is, for the time being)

   diff, v16, current

6. • CommentRowNumber6.
   • CommentAuthorAli Caglayan
   • CommentTimeFeb 11th 2019
   • (edited Feb 11th 2019)
   • PermaLink
   Author: Ali Caglayan
   Format: MarkdownItexIt is not strictly true that the hopf construction works for $S^0$ since it is not path-connected. However it is very easy to give the fibration explicitly. For the other spheres the hopf construction works fine. This is true straight out of Stasheff's book page 5. There are many places where this is incorrectly mentioned in [[Hopf construction]] and [[Hopf fibration]].

   It is not strictly true that the hopf construction works for $S^0$ since it is not path-connected. However it is very easy to give the fibration explicitly. For the other spheres the hopf construction works fine. This is true straight out of Stasheff’s book page 5. There are many places where this is incorrectly mentioned in Hopf construction and Hopf fibration.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 11th 2019
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   Author: Urs
   Format: MarkdownItexHm, I had another look at the Definition as stated in the entry [here](https://ncatlab.org/nlab/show/Hopf+construction#Definition) and plugged in $S^0 \simeq \mathbb{Z}/2$ for $X$. From, this I do seem to get the real Hopf fibratoipn $S^1 \overset{\cdot 2}{\longrightarrow} S^1$ all right. Let's go through it in detail: So on the left we have four copies of the interval, labeled in $\mathbb{Z}/2 \times \mathbb{Z}/2$, while on the right we have two copies, labeled by $\mathbb{Z}/2$. As we run once around on the left (with coordinates in $[0,1] \times (\mathbb{Z}_2 \times \mathbb{Z}_2) $) $$ (0,(0,0)) \to (1,(0,0)) \sim (1,(1,0)) \to (0,(1,0)) \sim (0,(1,1)) \to (1,(1,1)) \sim (1,(0,1)) \to (0,(0,1)) \sim (0,(0,0)) $$ we run on the right (with coordinates in $[0,1] \times \mathbb{Z}_2$) $$ (0,(0+0 = 0)) \to (1,(0 + 0 = 0)) \sim (1,(1 + 0 = 1)) \to (0,(1 + 0 = 1)) \sim (0,(1+1 = 0)) \to (1,(1+1 = 0)) \sim (1,(0 + 1 = 1)) \to (0,(0 + 1 = 1)) \sim (0,(0 + 0 = 0)) $$ That's twice around the circle on the right, for once on the left. Seems right to me. But let me know if I am missing something.

   Hm, I had another look at the Definition as stated in the entry here and plugged in $S^0 \simeq \mathbb{Z}/2$ for $X$. From, this I do seem to get the real Hopf fibratoipn $S^1 \overset{\cdot 2}{\longrightarrow} S^1$ all right.

   Let’s go through it in detail:

   So on the left we have four copies of the interval, labeled in $\mathbb{Z}/2 \times \mathbb{Z}/2$, while on the right we have two copies, labeled by $\mathbb{Z}/2$.

   As we run once around on the left (with coordinates in $[0,1] \times (\mathbb{Z}_2 \times \mathbb{Z}_2)$)

   $$(0,(0,0)) \to (1,(0,0)) \sim (1,(1,0)) \to (0,(1,0)) \sim (0,(1,1)) \to (1,(1,1)) \sim (1,(0,1)) \to (0,(0,1)) \sim (0,(0,0))$$

   we run on the right (with coordinates in $[0,1] \times \mathbb{Z}_2$)

   $$(0,(0+0 = 0)) \to (1,(0 + 0 = 0)) \sim (1,(1 + 0 = 1)) \to (0,(1 + 0 = 1)) \sim (0,(1+1 = 0)) \to (1,(1+1 = 0)) \sim (1,(0 + 1 = 1)) \to (0,(0 + 1 = 1)) \sim (0,(0 + 0 = 0))$$

   That’s twice around the circle on the right, for once on the left.

   Seems right to me. But let me know if I am missing something.

8. • CommentRowNumber8.
   • CommentAuthorAli Caglayan
   • CommentTimeFeb 11th 2019
   • (edited Feb 11th 2019)
   • PermaLink
   Author: Ali Caglayan
   Format: MarkdownItexI think the problem occurs because we get a quasifibration not necesserily a fibration. I am not so sure however.

   I think the problem occurs because we get a quasifibration not necesserily a fibration. I am not so sure however.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeFeb 11th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexSorry, which problem do you think occurs?

   Sorry, which problem do you think occurs?

10. • CommentRowNumber10.
    • CommentAuthorAli Caglayan
    • CommentTimeFeb 11th 2019
    • PermaLink
    Author: Ali Caglayan
    Format: MarkdownItexActually ignore what I said above that is complete nonsense. I found that X is 0-connected can be weakened to $\pi_0(X)$ is a group, so this is in fact correct. On page 4. The reason I got confused is because the multiplcation maps $X\times X \to X$ should be weak homotopy equivalences $X \simeq X$ when given one argument. It seems the book claims this can be proven on the weaker assumption that $\pi_0(X)$ is a group. Which means the articles are kind of correct, but only when this condition holds.

    Actually ignore what I said above that is complete nonsense. I found that X is 0-connected can be weakened to $\pi_0(X)$ is a group, so this is in fact correct. On page 4. The reason I got confused is because the multiplcation maps $X\times X \to X$ should be weak homotopy equivalences $X \simeq X$ when given one argument. It seems the book claims this can be proven on the weaker assumption that $\pi_0(X)$ is a group. Which means the articles are kind of correct, but only when this condition holds.

11. • CommentRowNumber11.
    • CommentAuthorAli Caglayan
    • CommentTimeFeb 11th 2019
    • PermaLink
    Author: Ali Caglayan
    Format: MarkdownItexIn general it is not correct to say the hopf construction works for any $X$ but for a $0$-connected $X$ or at least $\pi_0(X)$ being a group. When these are CW-complexes it seems quasifibrations become fibrations due to Dold. I am not so familar with proving left multiplication in a H-space is a weak homotopy equivalence however, especially with the connectedness condition weakened.

    In general it is not correct to say the hopf construction works for any $X$ but for a $0$-connected $X$ or at least $\pi_0(X)$ being a group. When these are CW-complexes it seems quasifibrations become fibrations due to Dold. I am not so familar with proving left multiplication in a H-space is a weak homotopy equivalence however, especially with the connectedness condition weakened.

12. • CommentRowNumber12.
    • CommentAuthorAli Caglayan
    • CommentTimeFeb 11th 2019
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    Author: Ali Caglayan
    Format: MarkdownItexI have [asked a question on MO](https://mathoverflow.net/questions/322962/does-the-hopf-construction-work-for-s0) hopefully we will get some correct answers rather than my vague thoughts.

    I have asked a question on MO hopefully we will get some correct answers rather than my vague thoughts.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItex$X$ must be an H-space. As it says in the entry.

    $X$ must be an H-space. As it says in the entry.

14. • CommentRowNumber14.
    • CommentAuthorAli Caglayan
    • CommentTimeFeb 11th 2019
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    Author: Ali Caglayan
    Format: MarkdownItex@Urs I am confused what you mean. Is it in the definition of H-space that left-multiplication is a homotopy equivalence? I thought a H-space was just a unital magma in the homotopy category.

    @Urs I am confused what you mean. Is it in the definition of H-space that left-multiplication is a homotopy equivalence? I thought a H-space was just a unital magma in the homotopy category.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded brief mentioning ([here](https://ncatlab.org/nlab/show/Hopf+fibration#RelationToStableHomotopyGroupsOfSpheres)) of the suspensions of the Hopf fibrations generating the corresponding stable homotopy groups of spheres <a href="https://ncatlab.org/nlab/revision/diff/Hopf+fibration/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hopf+fibration/17">v17</a>, <a href="https://ncatlab.org/nlab/show/Hopf+fibration">current</a>

    added brief mentioning (here) of the suspensions of the Hopf fibrations generating the corresponding stable homotopy groups of spheres

    diff, v17, current

16. • CommentRowNumber16.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 11th 2019
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    Author: Charles Rezk
    Format: MarkdownItexI don't see why the Hopf construction applied to an $H$-space $X$ must necessarily produce a quasifibration, even if we assume $\pi_0X$ is a group or even just $X$ connected. (It is certainly true that the *homotopy fiber* of $h\colon X\star X\to SX$ will be equivalent to $X$ under these hypotheses; I just don't see why the *actual fibers* of $h$ should also be equivalent to $X$, which is what necessarily must happen if $h$ is a quasifibration.)

    I don’t see why the Hopf construction applied to an $H$-space $X$ must necessarily produce a quasifibration, even if we assume $\pi_0X$ is a group or even just $X$ connected. (It is certainly true that the homotopy fiber of $h\colon X\star X\to SX$ will be equivalent to $X$ under these hypotheses; I just don’t see why the actual fibers of $h$ should also be equivalent to $X$, which is what necessarily must happen if $h$ is a quasifibration.)

17. • CommentRowNumber17.
    • CommentAuthorDylan Wilson
    • CommentTimeFeb 11th 2019
    • PermaLink
    Author: Dylan Wilson
    Format: Text@Charles: According to Stasheff (page 3), Sugawara proves this thing is a quasifibration for CW-H-spaces, and Dold-Lashof prove this is a quasifibration if m(x,-) is a weak equivalence for every x. But I haven't looked at the proofs- maybe you can find either reassurance or an error there?

    @Charles: According to Stasheff (page 3), Sugawara proves this thing is a quasifibration for CW-H-spaces, and Dold-Lashof prove this is a quasifibration if m(x,-) is a weak equivalence for every x. But I haven't looked at the proofs- maybe you can find either reassurance or an error there?
18. • CommentRowNumber18.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 11th 2019
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    Author: Charles Rezk
    Format: MarkdownItex@Dylan: I don't know where Sugawara's proof is found: there are three papers of his in the references of Stasheff's book, and I'm not going to hunt them all down. Suppose $X$ is a grouplike $H$-space, with multiplication $\mu$. Let $f\colon X\times X\to X$ be any morphism in the relative mapping space $\mathrm{Map}_{X\vee X/}(X\times X,X)$ in the path component of $\mu$. Every such $f$ determines a group-like $H$-space structure on $X$, and if we believe the theorem all fibers of all the Hopf constructions $h_f\colon X\star X\to SX$ must be weakly equivalent to $X$. The map $f$ can be found inside $h_f$ (i.e., $h_f$ restricts to $f$ along inclusions $X\times X\subset X\star X$ and $X\subset S(X)$), so in particular *all the fibers of all the maps $f$ over all points of $X$ must be weakly equivalent to $X$*. This seems unlikely. Dold-Lashof prove that a *different* construction gives a quasifibration; this other construction, applied to a group-like $H$-space, gives a map homotopic to the Hopf construction as usually defined.

    @Dylan: I don’t know where Sugawara’s proof is found: there are three papers of his in the references of Stasheff’s book, and I’m not going to hunt them all down.

    Suppose $X$ is a grouplike $H$-space, with multiplication $\mu$. Let $f\colon X\times X\to X$ be any morphism in the relative mapping space $\mathrm{Map}_{X\vee X/}(X\times X,X)$ in the path component of $\mu$. Every such $f$ determines a group-like $H$-space structure on $X$, and if we believe the theorem all fibers of all the Hopf constructions $h_f\colon X\star X\to SX$ must be weakly equivalent to $X$. The map $f$ can be found inside $h_f$ (i.e., $h_f$ restricts to $f$ along inclusions $X\times X\subset X\star X$ and $X\subset S(X)$), so in particular all the fibers of all the maps $f$ over all points of $X$ must be weakly equivalent to $X$. This seems unlikely.

    Dold-Lashof prove that a different construction gives a quasifibration; this other construction, applied to a group-like $H$-space, gives a map homotopic to the Hopf construction as usually defined.

19. • CommentRowNumber19.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 11th 2019
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    Author: Charles Rezk
    Format: MarkdownItexI'm guessing Stasheff is referring Theorem 4 (p. 119) of > Sugawara, Masahiro. > *On a condition that a space is an H-space.* > Math. J. Okayama Univ. 6 (1957), 109–129. This basically shows that the fiber of $h\colon X\star X\to SX$ over one of the cone points of $SX$ (which is in fact homeomorphic to $X$) is weakly equivalent (by the tautological map) to the homotopy fiber of $h$. It does *not* show that *every* fiber of $h$ is weakly equivalent (via the tautological map) to the corresponding homotopy fiber, so it does not show that $h$ is a quasifibration.

    I’m guessing Stasheff is referring Theorem 4 (p. 119) of

    Sugawara, Masahiro. On a condition that a space is an H-space. Math. J. Okayama Univ. 6 (1957), 109–129.

    This basically shows that the fiber of $h\colon X\star X\to SX$ over one of the cone points of $SX$ (which is in fact homeomorphic to $X$) is weakly equivalent (by the tautological map) to the homotopy fiber of $h$. It does not show that every fiber of $h$ is weakly equivalent (via the tautological map) to the corresponding homotopy fiber, so it does not show that $h$ is a quasifibration.

20. • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeFeb 15th 2019
    • (edited Feb 15th 2019)
    • PermaLink
    Author: Urs
    Format: MarkdownItexJust to say that I have recorded the simple example from [#7](#Comment_75971) (of working out the real Hopf fibration as the special case of the Hopf construction for $X = \mathbb{Z}/2$) to the entry *[[real Hopf fibration]]*: [here](https://ncatlab.org/nlab/show/real+Hopf+fibration#ViaJoinConstruction)

    Just to say that I have recorded the simple example from #7 (of working out the real Hopf fibration as the special case of the Hopf construction for $X = \mathbb{Z}/2$) to the entry real Hopf fibration: here

21. • CommentRowNumber21.
    • CommentAuthorAli Caglayan
    • CommentTimeFeb 15th 2019
    • PermaLink
    Author: Ali Caglayan
    Format: MarkdownItexThe reason I brought it up is because in HoTT we have "0-connectedness" floating around. I was further confused by Stasheff's book mentioning the condition. When I was learning Hopf fibrations in Algebraic topology I don't ever remember using the hopf construction to make them. I only really found out about it when learning HoTT. The mathoverflow question hasn't helped in the slightest, though I can't blame it since my question wasn't exactly well written. I am inclined to believe that it does work for $S^0$ though I am made slightly uncomfortable about what Charles has said. Now I know it isn't exactly a good motivation to think this way because of HoTT, but this still poses the question of why it is needed in HoTT. In the HoTT proof that the hopf construction gives a fibration, an essential part of which is showing left and right multiplication in the H-space is an equivalence. For this one "needs" 0-connectedness. Overall I couldn't care less if the hopf construction works for $S^0$, since the real hopf fibration is so easy to give explicitly in HoTT and AT. The only thing confusing me is the hopf construction working for $S^0$. And on this I am unfortunately still confused.

    The reason I brought it up is because in HoTT we have “0-connectedness” floating around. I was further confused by Stasheff’s book mentioning the condition. When I was learning Hopf fibrations in Algebraic topology I don’t ever remember using the hopf construction to make them. I only really found out about it when learning HoTT. The mathoverflow question hasn’t helped in the slightest, though I can’t blame it since my question wasn’t exactly well written. I am inclined to believe that it does work for $S^0$ though I am made slightly uncomfortable about what Charles has said.

    Now I know it isn’t exactly a good motivation to think this way because of HoTT, but this still poses the question of why it is needed in HoTT. In the HoTT proof that the hopf construction gives a fibration, an essential part of which is showing left and right multiplication in the H-space is an equivalence. For this one “needs” 0-connectedness.

    Overall I couldn’t care less if the hopf construction works for $S^0$, since the real hopf fibration is so easy to give explicitly in HoTT and AT. The only thing confusing me is the hopf construction working for $S^0$. And on this I am unfortunately still confused.

22. • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeFeb 15th 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have spelled it out twice now, in some detail [here](https://ncatlab.org/nlab/show/real+Hopf+fibration#ViaJoinConstruction). Both times you come back just re-iterating the question.

    I have spelled it out twice now, in some detail here. Both times you come back just re-iterating the question.

23. • CommentRowNumber23.
    • CommentAuthorAli Caglayan
    • CommentTimeFeb 15th 2019
    • (edited Feb 15th 2019)
    • PermaLink
    Author: Ali Caglayan
    Format: MarkdownItex@Urs and both time I agree, what I am confused about is why this doesn't work in HoTT. I am not at all disputing what you have written both times about $S^0$. The nlab article on [[Hopf construction]], attributes Theorem 3.1 to Sugawara. Unless I am mistaken, Charles has just said that Sugawara does not show this is a quasifibration, which is what Stasheff's book and the article claim. Is this not a problem?

    @Urs and both time I agree, what I am confused about is why this doesn’t work in HoTT. I am not at all disputing what you have written both times about $S^0$. The nlab article on Hopf construction, attributes Theorem 3.1 to Sugawara. Unless I am mistaken, Charles has just said that Sugawara does not show this is a quasifibration, which is what Stasheff’s book and the article claim. Is this not a problem?

24. • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeFeb 15th 2019
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    Author: Urs
    Format: MarkdownItexI may never have understood what your question actually is. You seemed to claim some problem for the case $S^0$, where I showed you that one gets the double cover map all right, which is as fibration as it gets.

    I may never have understood what your question actually is. You seemed to claim some problem for the case $S^0$, where I showed you that one gets the double cover map all right, which is as fibration as it gets.

25. • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeFeb 15th 2019
    • (edited Feb 15th 2019)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have made a brief edit to record Charles' observations from #18 and #19 [here](https://ncatlab.org/nlab/show/Hopf+construction#RealizationAsAQuasiFibration). No time for more.

    I have made a brief edit to record Charles’ observations from #18 and #19 here. No time for more.

26. • CommentRowNumber26.
    • CommentAuthorAli Caglayan
    • CommentTimeFeb 15th 2019
    • PermaLink
    Author: Ali Caglayan
    Format: MarkdownItex@Urs I have no problem with the case for $S^0$, perhaps my question is better phrased as "when is the hopf construction a fibration"? The problem I have for $S^0$ is that the article reads like the hopf construction gives you that it is a fibration, fortunately, as you said twice it is easy to see that this is actually a fibration for $S^0$. I just want to clear up the conditions needed for the hopf construction to give a fibration.

    @Urs I have no problem with the case for $S^0$, perhaps my question is better phrased as “when is the hopf construction a fibration”? The problem I have for $S^0$ is that the article reads like the hopf construction gives you that it is a fibration, fortunately, as you said twice it is easy to see that this is actually a fibration for $S^0$. I just want to clear up the conditions needed for the hopf construction to give a fibration.

27. • CommentRowNumber27.
    • CommentAuthorDylan Wilson
    • CommentTimeFeb 15th 2019
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    Author: Dylan Wilson
    Format: Text@Alizter: If what you really care about is HoTT, doesn't this not matter so much? You can use the Dold-Lashof result instead, which (as Charles explains) gives you the Hopf construction up to equivalence (which is invisible to HoTT) and identifies the homotopy fibers (indistinguishable from fibers in HoTT!) as what you want them to be.

    @Alizter: If what you really care about is HoTT, doesn't this not matter so much? You can use the Dold-Lashof result instead, which (as Charles explains) gives you the Hopf construction up to equivalence (which is invisible to HoTT) and identifies the homotopy fibers (indistinguishable from fibers in HoTT!) as what you want them to be.
28. • CommentRowNumber28.
    • CommentAuthorCharles Rezk
    • CommentTimeFeb 15th 2019
    • (edited Feb 15th 2019)
    • PermaLink
    Author: Charles Rezk
    Format: MarkdownItexActually, Sugawara proves everything you need: he shows that if $X$ is a *grouplike $H$-space" (i.e., a unital multiplication such that left multiplication by any element is an equivalence, and likewise for right multiplication), then the homotopy fiber of $h\colon X\star X\to S(X)$ over a particular point of $S(X)$ is equivalent to $X$. Since $S(X)$ must be connected whenever $X$ is non-empty, this means that the homotopy fiber over any point of $S(X)$ is equivalent to $X$. The HoTT proof is some sort of instantiation of the following observation: we have a map from the diagram \[ X \xleftarrow{\pi_1} X\times X \xrightarrow{\pi_2} X \] to the diagram \[ * \leftarrow X \rightarrow * \] where the map in the middle column $\mu\colon X\times X\to X$ is multiplication. The "grouplike" condition lets you prove that both commutative squares you get are homotopy pullbacks. Then "descent" applies to identify the fibers of the induced map from the homotopy colimit of the first diagram to the homotopy colimit of the second diagram. (Actually, the multiplication doesn't even need to be unital.)

    Actually, Sugawara proves everything you need: he shows that if $X$ is a *grouplike $H$-space” (i.e., a unital multiplication such that left multiplication by any element is an equivalence, and likewise for right multiplication), then the homotopy fiber of $h\colon X\star X\to S(X)$ over a particular point of $S(X)$ is equivalent to $X$. Since $S(X)$ must be connected whenever $X$ is non-empty, this means that the homotopy fiber over any point of $S(X)$ is equivalent to $X$.

    The HoTT proof is some sort of instantiation of the following observation: we have a map from the diagram

    $$X \xleftarrow{\pi_1} X\times X \xrightarrow{\pi_2} X$$

    to the diagram

    $$* \leftarrow X \rightarrow *$$

    where the map in the middle column $\mu\colon X\times X\to X$ is multiplication. The “grouplike” condition lets you prove that both commutative squares you get are homotopy pullbacks. Then “descent” applies to identify the fibers of the induced map from the homotopy colimit of the first diagram to the homotopy colimit of the second diagram. (Actually, the multiplication doesn’t even need to be unital.)

29. • CommentRowNumber29.
    • CommentAuthorAli Caglayan
    • CommentTimeFeb 17th 2019
    • PermaLink
    Author: Ali Caglayan
    Format: MarkdownItexThis makes much more sense now. Thank you @Urs, @Charles and @Dylan.

    This makes much more sense now. Thank you @Urs, @Charles and @Dylan.

30. • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeMar 18th 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded the following pointers: *** Discussion of [[supersymmetry|supersymmetric]] Hopf fibrations: * [[A. P. Balachandran]], G. Marmo, B.-S. Skagerstam and A. Stern, section 9.3 of _Gauge Symmetries and Fibre Bundles_, Lect. Notes in Physics 188, Springer-Verlag, Berlin, 1983 ([arXiv:1702.08910](https://arxiv.org/abs/1702.08910)) * Simon Davis, section 3 of _Supersymmetry and the Hopf fibration_ ([doi:10.4995/agt.2012.1623](https://doi.org/10.4995/agt.2012.1623)) <a href="https://ncatlab.org/nlab/revision/diff/Hopf+fibration/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hopf+fibration/19">v19</a>, <a href="https://ncatlab.org/nlab/show/Hopf+fibration">current</a>

    added the following pointers:

    ---

    Discussion of supersymmetric Hopf fibrations:

    • A. P. Balachandran, G. Marmo, B.-S. Skagerstam and A. Stern, section 9.3 of Gauge Symmetries and Fibre Bundles, Lect. Notes in Physics 188, Springer-Verlag, Berlin, 1983 (arXiv:1702.08910)

    • Simon Davis, section 3 of Supersymmetry and the Hopf fibration (doi:10.4995/agt.2012.1623)

    diff, v19, current

31. • CommentRowNumber31.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 29th 2019
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexAdded something on $Spin(3)$-equivariance. <a href="https://ncatlab.org/nlab/revision/diff/Hopf+fibration/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hopf+fibration/21">v21</a>, <a href="https://ncatlab.org/nlab/show/Hopf+fibration">current</a>

    Added something on $Spin(3)$-equivariance.

    diff, v21, current

32. • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2020
    • (edited Aug 5th 2020)
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to * Saifuddin Syed, _Group structure on spheres and the Hopf fibration_ ([pdf](https://ncatlab.org/nlab/files/SyedHopfFibration.pdf)) But the entry is lacking a canonical classical reference, for the definition via projective spaces... <a href="https://ncatlab.org/nlab/revision/diff/Hopf+fibration/25">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hopf+fibration/25">v25</a>, <a href="https://ncatlab.org/nlab/show/Hopf+fibration">current</a>

    added pointer to

    • Saifuddin Syed, Group structure on spheres and the Hopf fibration (pdf)

    But the entry is lacking a canonical classical reference, for the definition via projective spaces…

    diff, v25, current

33. • CommentRowNumber33.
    • CommentAuthorRichard Williamson
    • CommentTimeAug 5th 2020
    • (edited Aug 5th 2020)
    • PermaLink
    Author: Richard Williamson
    Format: MarkdownItex> But the entry is lacking a canonical classical reference, for the definition via projective spaces… If you mean a textbook reference (as opposed to the historical origin), it is Example 4.45 in Hatcher for example, which includes a nice picture.

    But the entry is lacking a canonical classical reference, for the definition via projective spaces…

    If you mean a textbook reference (as opposed to the historical origin), it is Example 4.45 in Hatcher for example, which includes a nice picture.

34. • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexI mean a decent reference on general Hopf fibrations via projective spaces. Some review that one can actually cite. Found one: Section 6 of Gluck-Warner-Yang [euclid.dmj/1077303489](https://projecteuclid.org/euclid.dmj/1077303489)

    I mean a decent reference on general Hopf fibrations via projective spaces. Some review that one can actually cite.

    Found one: Section 6 of Gluck-Warner-Yang euclid.dmj/1077303489

35. • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded these pointers: * [[Herman Gluck]], [[Frank Warner]], [[Chung Tao Yang]], Section 6 of: _Division algebras, fibrations of spheres by great spheres and the topological determination of space by the gross behavior of its geodesics_, Duke Math. J. Volume 50, Number 4 (1983), 1041-1076 ([euclid:dmj/1077303489](https://projecteuclid.org/euclid.dmj/1077303489)) * [[Herman Gluck]], [[Frank Warner]], [[Wolfgang Ziller]], _The geometry of the Hopf fibrations_, L'Enseignement Math&eacute;matique, t.32 (1986), p. 173-198 <a href="https://ncatlab.org/nlab/revision/diff/Hopf+fibration/26">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hopf+fibration/26">v26</a>, <a href="https://ncatlab.org/nlab/show/Hopf+fibration">current</a>

    added these pointers:

    • Herman Gluck, Frank Warner, Chung Tao Yang, Section 6 of: Division algebras, fibrations of spheres by great spheres and the topological determination of space by the gross behavior of its geodesics, Duke Math. J. Volume 50, Number 4 (1983), 1041-1076 (euclid:dmj/1077303489)

    • Herman Gluck, Frank Warner, Wolfgang Ziller, The geometry of the Hopf fibrations, L’Enseignement Mathématique, t.32 (1986), p. 173-198

    diff, v26, current

36. • CommentRowNumber36.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 20th 2021
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexUnified notation between complex and quaternionic descriptions (the complex formulation looks like it was copied from Wikipedia's page, with it's own notational style) <a href="https://ncatlab.org/nlab/revision/diff/Hopf+fibration/29">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hopf+fibration/29">v29</a>, <a href="https://ncatlab.org/nlab/show/Hopf+fibration">current</a>

    Unified notation between complex and quaternionic descriptions (the complex formulation looks like it was copied from Wikipedia’s page, with it’s own notational style)

    diff, v29, current

37. • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexthe original references were missing: * [[Heinz Hopf]], *Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche*, Mathematische Annalen **104** (1931) 637–665 &lbrack;[doi:10.1007/BF01457962](https://doi.org/10.1007/BF01457962)&rbrack; * [[Heinz Hopf]], *Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension*, Fundamenta Mathematicae **25** 1 (1935) 427-440 &lbrack;[eudml:212801](https://eudml.org/doc/212801)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Hopf+fibration/34">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hopf+fibration/34">v34</a>, <a href="https://ncatlab.org/nlab/show/Hopf+fibration">current</a>

    the original references were missing:

    • Heinz Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Mathematische Annalen 104 (1931) 637–665 [doi:10.1007/BF01457962]

    • Heinz Hopf, Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension, Fundamenta Mathematicae 25 1 (1935) 427-440 [eudml:212801]

    diff, v34, current